Question

You are given a rectangular piece of paper that has length x=19.7 cm and height y=18 cm. The lower right corner is to be folded to the top edge forming a triangle as shown. Determine the minimum area of a triangle that can be constructed.

Answer 1

To find the minimum area of a triangle formed by folding a paper, we would use the formula 1/2 × base × height. For a specific example with a base of 166 mm and height of 930.0 mm, the area of the triangle when converted to square meters is 0.0772 m², expressed to three significant figures.

The minimum area of a triangle that can be constructed by folding the lower right corner of a rectangular piece of paper to the top edge is achieved when the folded corner precisely reaches the top side of the rectangle. This forms a right-angle triangle whose dimensions are constrained by the rectangle's dimensions.

However, without loss or gain of material, the triangle's area would equal half of the rectangle's area. Since we know that the formula for the area of a triangle is 1/2 × base × height, we directly apply it. For example, if a triangle has a base of 166 mm and a height of 930.0 mm, its area in square meters after converting from millimeters is calculated as:

1. **Given Dimensions:**

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`Base (\(b\)) = 166 mm`

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`Height (\(h\)) = 930.0 mm`

2. **Area of the Triangle
`(\(A_{\text{triangle}}\))`:**

`\[ A_{\text{triangle}} = (1)/(2) * b * h \]`

3. **Substitute Given Values:**

`\[ A_{\text{triangle}} = (1)/(2) * 166 \, \text{mm} * 930.0 \, \text{mm} \]`

4. **Perform the Multiplication:**

`\[ A_{\text{triangle}} = (1)/(2) * 154,980 \, \text{mm}^2 \]`

5. **Convert to Square Meters:**

`\[ A_{\text{triangle}} = (1)/(2) * 0.15498 \, \text{m}^2 \]`

6. **Round to Three Significant Figures:**

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`\( A_{\text{triangle}} \approx 0.0772 \, \text{m}^2 \)`

When reporting the result, we should express our final answer to the correct number of significant figures, which in this case is three, making the area 0.0772 m².